log(log(x)) + log(-2 + log(x^3)) = 0
We know that log(a) + log(b) = log(ab), so lets apply that.
log(-2log(x) + log(x)(log(x^3)) = 0
If the base is supposedly 10 (of the logarithm), we can get rid of the outmost log on the right hand side by raising both sides to the power of 10, to get
-2log(x) + log(x)(log(x^3) = 10^0 = 1
We also know log(x^a) = alog(x), so lets apply that
-2log(x) + 3log(x)log(x) = 1, lets now let U = log(x)
-2U + 3U^2 = 1
3U^2 - 2U - 1 = 0,
(3U + 1)(U - 1) = 0
U = 1 or -1/3
log(x) = 1, so x = 10
log(x) = -1/3, 10^(-1/3) = x, however this is not a natural number, so x = 10